botanic curve

The botanic curve is the conchoid of the rhodonea.
Its name is derived from her petal-like form.

Other names for the curve are:

• geometric petal
• rose of Troy

The following qualities of the rhodonea hold also for the botanic curve:

• the number of petals is the denominator of 1/2 - 1/(2c)
• for integer values for c the number of petals is c (c odd), or 2c (c even)
• when c is irrational the curve does not close, and the number of petals is infinite
• when the parameter c is rational, the resulting curve is algebraic

The following botanic curves have been given a special name:

Some examples for other values of c:

You can observe that the value of d defines the form of the petals: for d < 1 the petal is open, it closes for d=1 and it forms an extra small petal for d>1.

John Baines made a Flash script to produce the curve.